Bayesian Quantile Regression with Adaptive MCMC

Abstract

Quantile regression offers a powerful means of characterizing how covariate effects vary across the entire outcome distribution, but standard implementations can suffer from curve crossings or computational burdens.  This study proposes an approach that builds conditional quantile curves sequentially—starting from the median and expanding outward—with constrained priors to enforce non-crossing by construction.  Posterior inference is carried out via an adaptive Metropolis algorithm, eliminating the need for closed-form full conditionals and improving mixing as the posterior concentrates.  Here we report a subset of results—single-predictor simulations under normal, right-skewed Gamma, and heteroscedastic errors across varying sample sizes. Results showed that proposed approach consistently attains lower bias and RMSE than both frequentist fits and Gibbs quantile regression, and achieves superior convergence efficiency.  An empirical application to the 2023 Philippine FIES—modeling log educational spending on log household income—demonstrates the proposed approach's ability to produce coherent, non-crossing quantile estimates that uncover increasing income elasticities across spending levels.  These results highlight its practical utility for distributional analysis where monotonicity and computational efficiency are essential.